assumes (at adiabatic-isochoric acceleration) the value as given by equation (6).
This theorem can be easier derived in the following way (even though it is less clear from the physical standpoint): At adiabatic state changes, the amount of only depends on the momentary values of the quantities and . Thus when (at arbitrary velocity) is adiabatically changed by , then changes by .[1] Then the energy increase, which is equal to the work of the external forces here, is:
and therefore also
is a complete differential, thus
Here, is to be understood as a differentiation at adiabatic state change; thus if is given as an explicit function of and , we have:
Furthermore, since we have according to (4)
the previous equation can be integrated towards and we obtain
This constant can also be a function of and ; it reduces to zero, since we have